If you look up at that circle thing above the elevator, you can see streaking black vertical lines going across it. This happens on some textures in the game when Logarithmic is disabled. Streaking black vertical lines will be rendered at a distance, instead of the correct objects.

This problem happens on all versions of PCSX2 that FFXII works on (0.9.5, and the newly released playground version of PCX2). I've also narrowed it down to it being a Logarithmic Z issue only, as well. All speed hack options were off, and I turned on all GSDX options one at a time to see what was causing the problem.

Is this a common issue with Logarithmic Z and FFXII? Or is there something I can do to get Logarithmic Z to work properly?

Your problem cannot be solved easily. To get it working correctly you probably need to use directx 10.
That means Windows Vista and a dx10 capable graphic card.
Alternatively you can try running the game in gsdx software mode. At least to see if that fixes it.